\(\int \frac {(a+b \text {arccosh}(c x))^2}{(\pi -c^2 \pi x^2)^{5/2}} \, dx\) [49]

Optimal result

Integrand size = 26, antiderivative size = 289 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 \pi ^{5/2} \sqrt {1-c x} \sqrt {1+c x}}+\frac {b \sqrt {-1+c x} (a+b \text {arccosh}(c x))}{3 c \pi ^{5/2} \sqrt {1-c x} \left (1-c^2 x^2\right )}+\frac {2 \sqrt {-1+c x} (a+b \text {arccosh}(c x))^2}{3 c \pi ^{5/2} \sqrt {1-c x}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -c^2 \pi x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 \pi ^2 \sqrt {\pi -c^2 \pi x^2}}-\frac {4 b \sqrt {-1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 c \pi ^{5/2} \sqrt {1-c x}}-\frac {2 b^2 \sqrt {-1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 c \pi ^{5/2} \sqrt {1-c x}} \] Output:

-1/3*b^2*x/Pi^(5/2)/(-c*x+1)^(1/2)/(c*x+1)^(1/2)+1/3*b*(c*x-1)^(1/2)*(a+b* 
arccosh(c*x))/c/Pi^(5/2)/(-c*x+1)^(1/2)/(-c^2*x^2+1)+2/3*(c*x-1)^(1/2)*(a+ 
b*arccosh(c*x))^2/c/Pi^(5/2)/(-c*x+1)^(1/2)+1/3*x*(a+b*arccosh(c*x))^2/Pi/ 
(-Pi*c^2*x^2+Pi)^(3/2)+2/3*x*(a+b*arccosh(c*x))^2/Pi^2/(-Pi*c^2*x^2+Pi)^(1 
/2)-4/3*b*(c*x-1)^(1/2)*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1) 
^(1/2))^2)/c/Pi^(5/2)/(-c*x+1)^(1/2)-2/3*b^2*(c*x-1)^(1/2)*polylog(2,(c*x+ 
(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/c/Pi^(5/2)/(-c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.97 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-a^2 c x \left (-3+2 c^2 x^2\right )+a b \left (-2 c x \left (-3+2 c^2 x^2\right ) \text {arccosh}(c x)+\frac {\left (-1+c^2 x^2\right ) \left (1+2 \left (-1+c^2 x^2\right ) \log (-1+c x)+2 \left (-1+c^2 x^2\right ) \log (1+c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right )+b^2 \left (1-c^2 x^2\right ) \left (-\frac {\text {arccosh}(c x) \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+c x \text {arccosh}(c x)\right )}{-1+c^2 x^2}+c x \left (-1+2 \text {arccosh}(c x)^2\right )-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1-e^{-2 \text {arccosh}(c x)}\right )\right )+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )\right )}{3 c \pi ^{5/2} \left (1-c^2 x^2\right )^{3/2}} \] Input:

Integrate[(a + b*ArcCosh[c*x])^2/(Pi - c^2*Pi*x^2)^(5/2),x]
 

Output:

(-(a^2*c*x*(-3 + 2*c^2*x^2)) + a*b*(-2*c*x*(-3 + 2*c^2*x^2)*ArcCosh[c*x] + 
 ((-1 + c^2*x^2)*(1 + 2*(-1 + c^2*x^2)*Log[-1 + c*x] + 2*(-1 + c^2*x^2)*Lo 
g[1 + c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + b^2*(1 - c^2*x^2)*(-((ArcCo 
sh[c*x]*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + c*x*ArcCosh[c*x]))/(-1 + c 
^2*x^2)) + c*x*(-1 + 2*ArcCosh[c*x]^2) - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + 
 c*x)*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 - E^(-2*ArcCosh[c*x])]) + 2*Sqr 
t[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*c*P 
i^(5/2)*(1 - c^2*x^2)^(3/2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.91 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6316, 6314, 6327, 6328, 3042, 26, 4199, 25, 2620, 2715, 2838, 6329, 41}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -\pi c^2 x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6316

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}}dx}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 6314

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 6327

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 6328

\(\displaystyle \frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 4199

\(\displaystyle \frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 6329

\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c}+\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 41

\(\displaystyle \frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{\pi c \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \text {arccosh}(c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}+\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 \pi ^2 \sqrt {\pi -\pi c^2 x^2}}\)

Input:

Int[(a + b*ArcCosh[c*x])^2/(Pi - c^2*Pi*x^2)^(5/2),x]
 

Output:

(x*(a + b*ArcCosh[c*x])^2)/(3*Pi*(Pi - c^2*Pi*x^2)^(3/2)) + (2*b*c*Sqrt[-1 
 + c*x]*Sqrt[1 + c*x]*(-1/2*(b*x)/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + 
b*ArcCosh[c*x])/(2*c^2*(1 - c^2*x^2))))/(3*Pi^2*Sqrt[Pi - c^2*Pi*x^2]) + ( 
2*((x*(a + b*ArcCosh[c*x])^2)/(Pi*Sqrt[Pi - c^2*Pi*x^2]) + ((2*I)*b*Sqrt[- 
1 + c*x]*Sqrt[1 + c*x]*(((-1/2*I)*(a + b*ArcCosh[c*x])^2)/b - (2*I)*(-1/2* 
((a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])]) - (b*PolyLog[2, E^(2*Ar 
cCosh[c*x])])/4)))/(c*Pi*Sqrt[Pi - c^2*Pi*x^2])))/(3*Pi)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 41

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S 
imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[b*c + a*d, 0]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6314

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), 
x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]   Int[x*((a 
 + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, 
 x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
 

rule 6316

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + 
 c*x)^p*(-1 + c*x)^p)]   Int[x*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a 
+ b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
 

rule 6327

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( 
e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 
*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
 

rule 6328

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[1/e   Subst[Int[(a + b*x)^n*Coth[x], x], x, ArcCosh[c*x] 
], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 6329

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + 
 c*x)^p)]   Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x 
])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && 
GtQ[n, 0] && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2372\) vs. \(2(267)=534\).

Time = 0.36 (sec) , antiderivative size = 2373, normalized size of antiderivative = 8.21

method	result	size
default	\(\text {Expression too large to display}\)	\(2373\)
parts	\(\text {Expression too large to display}\)	\(2373\)

Input:

int((a+b*arccosh(c*x))^2/(-Pi*c^2*x^2+Pi)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

4/3*b^2/Pi^(5/2)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2 
-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+4/3*b^2/Pi^(5/2)*(c 
*x+1)^(1/2)*(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1)*arccosh(c*x)*ln 
(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+4/3*b^2*(-c^2*x^2+1)^(1/2)/Pi^(5/2)/(3 
*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*(c*x-1)*(c*x+1)*arccosh(c*x)*x^5+2*b 
^2*(-c^2*x^2+1)^(1/2)/Pi^(5/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^3*(c* 
x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2*x^4-10/3*b^2*(-c^2*x^2+1)^(1/2)/Pi 
^(5/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2*(c*x-1)*(c*x+1)*arccosh(c*x 
)*x^3-14/3*b^2*(-c^2*x^2+1)^(1/2)/Pi^(5/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^ 
2-4)*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2*x^2+b^2*(-c^2*x^2+1)^(1/ 
2)/Pi^(5/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c*(c*x-1)^(1/2)*(c*x+1)^(1 
/2)*arccosh(c*x)*x^2+a^2*(1/3/Pi*x/(-Pi*c^2*x^2+Pi)^(3/2)+2/3/Pi^2*x/(-Pi* 
c^2*x^2+Pi)^(1/2))-16/3*b^2*(-c^2*x^2+1)^(1/2)/Pi^(5/2)/(3*c^6*x^6-10*c^4* 
x^4+11*c^2*x^2-4)*c^2*arccosh(c*x)*x^3-4/3*b^2*(-c^2*x^2+1)^(1/2)/Pi^(5/2) 
/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2/3*b^2 
*(-c^2*x^2+1)^(1/2)/Pi^(5/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*(c*x-1)*( 
c*x+1)*x-4/3*b^2*(-c^2*x^2+1)^(1/2)/Pi^(5/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2* 
x^2-4)*c^6*arccosh(c*x)*x^7-2*b^2*(-c^2*x^2+1)^(1/2)/Pi^(5/2)/(3*c^6*x^6-1 
0*c^4*x^4+11*c^2*x^2-4)*c^4*arccosh(c*x)^2*x^5+14/3*b^2*(-c^2*x^2+1)^(1/2) 
/Pi^(5/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4*arccosh(c*x)*x^5+17/3...
 

Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi - \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))^2/(-pi*c^2*x^2+pi)^(5/2),x, algorithm="fricas 
")
 

Output:

integral(-sqrt(pi - pi*c^2*x^2)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + 
 a^2)/(pi^3*c^6*x^6 - 3*pi^3*c^4*x^4 + 3*pi^3*c^2*x^2 - pi^3), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{4} \sqrt {- c^{2} x^{2} + 1} - 2 c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx + \int \frac {b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{c^{4} x^{4} \sqrt {- c^{2} x^{2} + 1} - 2 c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx + \int \frac {2 a b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {- c^{2} x^{2} + 1} - 2 c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \] Input:

integrate((a+b*acosh(c*x))**2/(-pi*c**2*x**2+pi)**(5/2),x)
 

Output:

(Integral(a**2/(c**4*x**4*sqrt(-c**2*x**2 + 1) - 2*c**2*x**2*sqrt(-c**2*x* 
*2 + 1) + sqrt(-c**2*x**2 + 1)), x) + Integral(b**2*acosh(c*x)**2/(c**4*x* 
*4*sqrt(-c**2*x**2 + 1) - 2*c**2*x**2*sqrt(-c**2*x**2 + 1) + sqrt(-c**2*x* 
*2 + 1)), x) + Integral(2*a*b*acosh(c*x)/(c**4*x**4*sqrt(-c**2*x**2 + 1) - 
 2*c**2*x**2*sqrt(-c**2*x**2 + 1) + sqrt(-c**2*x**2 + 1)), x))/pi**(5/2)
 

Maxima [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi - \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))^2/(-pi*c^2*x^2+pi)^(5/2),x, algorithm="maxima 
")
 

Output:

1/3*a*b*c*(sqrt(-pi)/(pi^3*c^4*x^2 - pi^3*c^2) + 2*sqrt(-pi)*log(c*x + 1)/ 
(pi^3*c^2) + 2*sqrt(-pi)*log(c*x - 1)/(pi^3*c^2)) + 2/3*a*b*(x/(pi*(pi - p 
i*c^2*x^2)^(3/2)) + 2*x/(pi^2*sqrt(pi - pi*c^2*x^2)))*arccosh(c*x) + 1/3*a 
^2*(x/(pi*(pi - pi*c^2*x^2)^(3/2)) + 2*x/(pi^2*sqrt(pi - pi*c^2*x^2))) + b 
^2*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(pi - pi*c^2*x^2)^(5 
/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arccosh(c*x))^2/(-pi*c^2*x^2+pi)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (\Pi -\Pi \,c^2\,x^2\right )}^{5/2}} \,d x \] Input:

int((a + b*acosh(c*x))^2/(Pi - Pi*c^2*x^2)^(5/2),x)
 

Output:

int((a + b*acosh(c*x))^2/(Pi - Pi*c^2*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\frac {6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{2} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b +3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2}+2 a^{2} c^{2} x^{3}-3 a^{2} x}{3 \sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \pi ^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int((a+b*acosh(c*x))^2/(-Pi*c^2*x^2+Pi)^(5/2),x)
 

Output:

(6*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 
 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*a*b*c** 
2*x**2 - 6*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c 
**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x) 
*a*b + 3*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)**2/(sqrt( - c**2*x**2 + 1)* 
c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x 
)*b**2*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)**2/(sqrt( - c** 
2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2* 
x**2 + 1)),x)*b**2 + 2*a**2*c**2*x**3 - 3*a**2*x)/(3*sqrt(pi)*sqrt( - c**2 
*x**2 + 1)*pi**2*(c**2*x**2 - 1))